Fungsi Kuadrat Bagian 1 - Matematika Wajib Kelas X m4thlab

Introduction

In this tutorial, we will explore the fundamentals of quadratic functions, including their general form, graphical representation, and key characteristics such as intercepts, vertex, and symmetry. This guide is designed for students in Class X who are looking to grasp the essential concepts of quadratic functions in mathematics.

Step 1: Understanding Parabolas

• A quadratic function represents a parabola, which can open upwards or downwards.

• The general form of a quadratic function is:

  [ f(x) = ax^2 + bx + c ]

  where:

  • (a) determines the direction of the parabola (upward if (a > 0), downward if (a < 0)).
  • (b) and (c) affect the position of the graph.

Step 2: Identifying the General Form of Quadratic Functions

• Recognize the components of the quadratic function:
  • (a): coefficient of (x^2)
  • (b): coefficient of (x)
  • (c): constant term
• Example: In (f(x) = 2x^2 + 3x + 1), (a = 2), (b = 3), and (c = 1).

Step 3: Graphing the Quadratic Function

• Plot the graph of the quadratic function based on its general form.
• Key points to consider:
  • The shape of the parabola.
  • How the coefficients (a), (b), and (c) affect the graph.

Step 4: Finding X-Intercepts

• To find the points where the graph intersects the X-axis, set (f(x) = 0):

  [ ax^2 + bx + c = 0 ]

• Use the quadratic formula:

  [ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ]

• The solutions will give you the X-intercepts.

Step 5: Finding Y-Intercept

• To find the Y-intercept, substitute (x = 0) into the function:

  [ f(0) = c ]

• The Y-intercept is the point ((0, c)) on the graph.

Step 6: Determining the Vertex

• The vertex (or turning point) can be found using the formula:

  [ x = -\frac{b}{2a} ]

• Calculate (y) by substituting the (x) value back into the function.

Step 7: Finding the Axis of Symmetry

• The axis of symmetry is a vertical line that passes through the vertex:

  [ x = -\frac{b}{2a} ]

Step 8: Identifying Maximum or Minimum Values

• If (a > 0), the vertex represents a minimum value.
• If (a < 0), it represents a maximum value.
• The y-coordinate of the vertex is the maximum or minimum value of the function.

Step 9: Determining Domain and Range

• Domain: The set of all possible (x) values. For quadratic functions, the domain is all real numbers: ((-∞, +∞)).
• Range:
  • If (a > 0): The range is ([k, +∞)) where (k) is the y-coordinate of the vertex (minimum).
  • If (a < 0): The range is ((-∞, k]) where (k) is the y-coordinate of the vertex (maximum).

Conclusion

In this tutorial, we've covered the basics of quadratic functions, including their general form, how to find intercepts, the vertex, axis of symmetry, and understanding domain and range. These concepts are crucial for solving quadratic equations and graphing parabolas. For further practice, explore additional examples or watch related tutorials to deepen your understanding.